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Abstract

Background

DNA methylation plays a very important role in the silencing of tumor suppressor genes
in various tumor types. In order to gain a genome-wide understanding of how changes
in methylation affect tumor growth, the differential methylation hybridization (DMH)
protocol has been developed and large amounts of DMH microarray data have been generated.
However, it is still unclear how to preprocess this type of microarray data and how
different background correction and normalization methods used for two-color gene
expression arrays perform for the methylation microarray data. In this paper, we demonstrate
our discovery of a set of internal control probes that have log ratios (M) theoretically
equal to zero according to this DMH protocol. With the aid of this set of control
probes, we propose two LOESS (or LOWESS, locally weighted scatter-plot smoothing)
normalization methods that are novel and unique for DMH microarray data. Combining
with other normalization methods (global LOESS and no normalization), we compare four
normalization methods. In addition, we compare five different background correction
methods.

Results

We study 20 different preprocessing methods, which are the combination of five background
correction methods and four normalization methods. In order to compare these 20 methods,
we evaluate their performance of identifying known methylated and un-methylated housekeeping
genes based on two statistics. Comparison details are illustrated using breast cancer
cell line and ovarian cancer patient methylation microarray data. Our comparison results
show that different background correction methods perform similarly; however, four
normalization methods perform very differently. In particular, all three different
LOESS normalization methods perform better than the one without any normalization.

Conclusions

It is necessary to do within-array normalization, and the two LOESS normalization
methods based on specific DMH internal control probes produce more stable and relatively
better results than the global LOESS normalization method.

Background

Microarray technology has been used extensively in genetic and epigenetic studies
over the last ten years. Several microarray platforms are available including the
single-channel Affymetrix oligonucleotide arrays, the two-color (or two-channel) cDNA
arrays, and Agilent two color arrays. In the two-color use, which is the focus of
this paper, two samples (or target genes) are labeled using two different fluorophores
(usually a red fluorescent dye, Cy5, and a green fluorescent dye, Cy3) and hybridized
simultaneously onto each probe (or spot) of the array (or chip). Then the arrays are
laser-scanned and images are processed to obtain the data for analysis [1]. In general, the log ratio Cy5 over Cy3 at each probe is used as a measurement. With
this microarray technology, studying thousands of genes simultaneously becomes possible.
For example, gene expression, copy number variation, and methylation patterns have
been widely studied using microarray technologies. However, due to some experimental
artifacts, random noise and systematic variation do exist in such high throughput
microarray experiments. Therefore, preprocessing, such as background correction and
normalization, is important to eliminate technical bias in order to identify real
biological variations.

Preprocessing gene expression microarray data obtained from two-color cDNA microarray
and single-color Affymetrix array have been extensively studied [2-4]. However, two-color methylation microarrays, especially the DNA methylation microarrays
generated based on the DMH protocol [5-7] with the Agilent technology [8], have not been well studied. DNA methylation arrays are very different from gene
expression arrays. The differences mainly lie in the following two aspects. First,
different materials are hybridized onto the array. For the gene expression array,
it is the mRNA that is reverse transcribed to cDNA. While for the DNA methylation
microarray, it is the DNA fragments selected based on the methylation-sensitive restriction
enzymes (MSRE) [5,7,9-11] or methyl-cytosine-specific antibody [12-15]. Second, they measure different biological phenomena, one measures gene expression,
or the mRNA levels, and another measures methylation signals. It has been recognized
that preprocessing methods for microarrays are platform specific and challenging to
automate [2,16]. It is still unknown whether gene expression array preprocessing methods can be applied
to the Agilent two-color methylation microarray. If applied, it is not clear how different
background correction and normalization methods would perform.

After the image analysis, foreground and background intensities are estimated for
each probe (or spot), and these intensities are usually denoted as Rf, Gf, Rb, and Gb respectively for the two channels (i.e., the red and green channels). The foreground
estimates (Rf and Gf) are the overall measurement of the intensity at each probe (spot) in each channel.
The background measurements (Rb, Gb) are usually an estimate of the ambient signal around the round circle of each spot.
This may be due to unequal distribution of hybridization solution, spatial bias [17], non-specific binding of labeled samples to the array surface, or non-hybridized
DNA not washed away [18]. Removing these ambient signals around each probe and adjusting the foreground signals
accordingly is called background correction.

The traditional background correction for gene expression microarray data is to subtract
the background estimate from the foreground intensity. This may produce negative intensities
and lead to missing log ratios. Log ratios are highly variable at low intensity probes.
In order to avoid this problem, three strategies that have been proposed are summarized
in [18]: 1) avoid background correction [2,19]; 2) use a different image analysis software to produce new background estimates,
for example, the 'morph' background measurement used in the Spot software (CSIRO,
North Ryde, Australia) or the TV+L model proposed by Yin et al. [20]; and 3) use statistical models to adjust the background estimate [17,18,21,22]. With these three strategies, there are eight different background methods and they
have been compared using the two color cDNA gene expression array data as shown in
Ritchie et al. [18]. According to this comparison, the standard background correction (i.e., subtraction)
performs far worse than other alternatives in that it produces a larger number of
false discoveries. Ritchie et al. [18] also shows that variance stabilization methods perform best, especially the 'normexp+offset'
method, which gives the lowest false discovery rate. Whether these conclusions are
valid in the methylation data generated by the Agilent technology is still unknown.
In addition, Ritchie et al. [18] only compare background correction methods and do not demonstrate how normalization
methods will affect and interact with different background correction methods.

In order to obtain accurate measurements from microarray technology, we must consider
the random and systematic variation due to some experimental artifacts. Normalization
of microarray data is the process of removing or adjusting these systematic biases
that usually include intensity dependent bias, dye bias and spatial effects [2,4]. A commonly used normalization method is the intensity dependent LOESS normalization
that fits a locally weighted polynomial regression to the average of the red and green
intensities, that is, the LOESS curve [2,4]. This LOESS normalization generally involves two steps [23]: (1) select probes (or genes) used to do normalization, and (2) apply a LOESS or
weighted LOESS to the data. The probes or genes that are normally selected are all
probes (genes), the housekeeping genes, the spike-in control probes, and microarray
sample pool control (MSP). Housekeeping genes have originally been used for normalization
because they are believed to have stable function and stable gene expression values.
However, it has been shown that they have large variability between different samples
and treatments [24,25]. Spike-in controls may be more trustworthy, but not all microarray experiments have
included spike-in controls. Microarray sample pool controls [2] are designed for gene expression data normalization, and their performance is still
unknown for methylation data. To the best of our knowledge, no probes or genes are
selected specifically for normalizing DMH microarray data.

In this paper, we demonstrate the identification of a set of probes that are specially
selected as internal control probes for the DMH protocol. Utilizing these DMH internal
control probes, we propose two LOESS normalization methods that are novel and unique
for the DMH methylation microarray data. Combining these two control probe LOESS methods
with other two standard normalization methods (global LOESS normalization and no normalization),
we compare four normalization methods. In addition, we also compare five different
background correction methods. Combining all these different background correction
and normalization methods results in 20 different preprocessing methods for the DNA
methylation microarrays. In order to see which preprocessing method can best identify
known methylated and housekeeping genes, all 20 methods are compared using microarray
data generated from breast cancer cell lines and ovarian cancer patients.

Results

DMH microarray protocol and data sets

Microarray technologies have revolutionized our understanding of genetics and epigenetics
at molecular levels. In particular, they have made it possible to identify DNA methylation
(a type of epigenetic modification) patterns simultaneously in many specific regions
or even the whole genome. The differential methylation hybridization (DMH) protocol
[5-7] is capable of evaluating the methylation pattern of all CpG islands in the whole
genome. This assay includes the following three steps. More details of the description
of the DMH protocols can be found in the literature [5-7,26].

1) Sonicating DNA sequences into 400-500 bp fragments, and then ligating these fragments
using linkers.

2) Digesting ligated DNA fragments using two MSREs, HpaII and HinpI, which have the
recognition cutting sites CCGG and CGCG respectively. If a DNA fragment contains at
least one recognition cutting site that is not methylated, it will be restricted (i.e.,
cut), and will not be hybridized onto the microarray. Therefore, it does not contribute
to the final methylation signals.

3) Using the polymerase chain reaction (PCR) to amplify the unrestricted DNA fragments
and then hybridizing them onto microarrays.

The above three steps are done for both test (cancer patients or cell lines) and control
(common normal references) samples. Then both samples are hybridized to the array
coupled with red or green fluorescent dyes. Here we use the Agilent 244K arrays hybridized
with the test samples (e.g., cancer cell lines) labeled with Cy5 (red dye, or R) and
a common normal reference labeled with Cy3 (green dye, or G). Two color arrays produce
both foreground, i.e., Rf and Gf, and background, i.e., Rb and Gb, intensities. After some proper background correction and normalization based on
Rf, Gf, Rb and Gb, we obtain the true signals R and G. We use the base two log ratio of red over green
intensity, log2(R/G), as the observed methylation signal at each probe. This is called the M value,
that is, log2(R) - log2(G). The average is (log2(R) + log2(G))/2 and is called the A value. The MA plot (with A values in the x-axis and M values
in the y-axis) is often used to examine dye bias before doing any normalization.

In this paper, we study 20 different preprocessing methods that are the combination
of five-background correction and four normalization methods. These comparisons are
done using two microarray data sets from 40 breast cancer cell lines and 26 ovarian
cancer patients. For each array in these two data sets, we preprocess it with different
background correction and normalization methods and then examine which preprocessing
method is better at identifying known methylated and non-methylated genes. For the
breast cancer cell line data, 30 known methylated genes [27-30] are used. For the ovarian cancer data, 32 known methylated genes are selected [31]. For the non-methylated genes, we use 47 known housekeeping genes selected from publicly
available data [32].

Review of background correction methods

1) None: no background correction and simply let R = Rf and G = Gf.

2) Subtract: this is the traditional background correction method with the local background
estimate subtracted from the foreground estimate. That is, R = Rf - Rb and G = Gf - Gb.

3) Edwards: in order to avoid the situation of local background estimates less than
foreground estimates, Edwards [17] proposes to subtract background (Rb and Gb) from foreground (Rf and Gf) when their difference is larger than a certain threshold d, otherwise, replace the subtraction by a smooth monotonic function. The detailed
formula is given as follows:

4) Normexp: this method applies a normal-exponential (i.e., normexp) convolution model
to the local background and the true signal at the red and green channels separately
[18,22]. For example, at the red channel, let S be the unknown true signal, let B be the
background noise that is not included in Rb, and let X = Rf-Rb be the background-corrected observed intensity. According to the normexp model, S
~ exp(a) (i.e., an exponential distribution with mean a), B ~N(μ,σ2) (i.e., a normal distribution with mean μ and variance σ2), and S and B are independent and additive. Therefore, we have X = S + B. We can
derive the intensity function for S and X, and then the conditional density of S|X.
The estimate of the unknown true signal S is the conditional expectation E(S|X = x).
The three key parameters, a, μ and σ2, can be estimated using a saddle-point approximation or the maximum likelihood method
[18,22]. The true signals in red and green channels, which are usually denoted as R and G,
can be obtained and their log ratio, log2(R/G), will be used as the methylation signal
at each probe.

5) Normexp+offset: this is the same as the Normexp method except that a small positive
offset is added to both channels to reduce the variance of low intensity log ratio
values. That is, the new log ratio value is equal to log2[(R+k)/(G+k)]. As used in
[18], we let k = 50.

Novel and existing normalization methods

The basic rationale of normalization is to remove or adjust for artifacts caused by
microarray technology rather than biological differences of the samples between printed
probes. In order to do so, it is helpful to normalize the data with some known information,
for example, some control probes that are known to be non-differentially methylated.
In this section, we first identify the probes that are known to be not differentially
methylated based on the DMH protocol, that is, probes with M = 0. DNA fragments are
restricted by two MSREs, HinpI and HapII, which have the recognition cutting sites
CGCG and CCGG respectively. If a DNA fragment contains at least one cutting site that
is not methylated, it will be restricted (i.e., cut), and will not be hybridized onto
the microarray. If a DNA fragment does not have any cutting sites, it will not be
digested by any MSREs and can be hybridized onto the array. If all the cutting sites
of a DNA fragment are methylated, this fragment will be saved for hybridization onto
the array. These three typical types of DNA fragments with examples are given in Table
1.

If there is not any recognition cutting site within a long region around a probe,
the hybridization from two channels are supposed to be the same whether or not there
is methylation on the DNA fragment that is hybridized onto this probe. Therefore,
theoretically the log 2 ratio of this probe methylation signal will be 0 (i.e., M
= 0). In this paper, we identified the probes around which there are no recognition
cutting sites within L = 900 base region. These probes are selected in the following
way. For each probe, we check the regions that are L = 900 bases around the center
of each probe. That is, there are L/2 bases on each side of the center of the probe.
Then we check how many restriction cutting sites are around this probe within these
L bases. If there are no cutting sites (i.e., the sequences CGCG and CCGG), we claim
that this probe is a non-differential methylated internal control probe with M = 0.
These internal control probes are important because we can make full use of them to
do normalizations. Because the length of DNA fragments is about 400-500 bp, we use
L = 900 bases assuming that DNA fragments can be hybridized onto the methylated probes
and regions evenly. 199 probes, which have no recognition cutting sites around 900
bp, are identified.

Similar to the LOESS normalization and composite LOESS normalization using control
probes in the context of the gene expression microarray preprocessing, we introduce
the control and composite LOESS normalization for DMH methylation microarray data.
Combining with the standard LOESS normalization and the method without any normalization,
there are four normalization methods. Details about these four normalization methods
are described as below:

3) Control LOESS normalization: We fit a LOESS curve only using the 199 control probes,
and for each M value which corresponds to an A value, we have Mnew = M- fcontrol(A).

4) Composite LOESS normalization: This is to let the normalization curve to be a weighted
average of the global LOESS curve and the control probe LOESS curve. That is, at each
specific average intensity level A, the new normalized estimate is g(A) = a* fcontrol(A) + (1-a) fall(A), and Mnew = Mobserved - g(A), where fall and fcontrol are the global and control LOESS curve, and 'a' is defined as the proportion of genes
less than a given intensity A value [2].

In Figure 1, we show an example of the MA plot of an array that is fitted with three different
LOESS curves: global LOESS (blue line), composite LOESS (cyan line), and control LOESS
(red line). This figure shows that there are some differences among these three LOESS
curves, so the normalization based on these three LOESS curves could be very different.
In this Figure, the red dots are the 199 internal control probes with M = 0 as their
theoretical log ratio values according to our DMH protocol. As we see that some probes
have some unexpected large and small log ratios, this could be due to some experimental
artefacts. Therefore, we should preprocess the raw microarray data first.

Figure 1.MA plot of one array with three LOESS curves. The blue line is the LOESS curve based on all biological probes. The red dots are
199 internal control probes. The red line is the LOESS curve obtained only using these
internal control probes. The cyan line is the weighted LOESS curve (i.e., composite
LOESS) curve based on both all biological probes and 199 internal control probes.

Comparison methods

All 20 different preprocessing methods as described are implemented using the LIMMA
package [4] of Bioconductor [33]. In order to compare the different background and normalization methods, we use the
quantile regression method [34], which identifies commonly hypermethylated genes in DMH microarray data. The basic
idea is that, for each CpG island we apply the quantile regression model [35] to the normalized M values obtained from 20 different preprocessing methods. Then
we use some known methylated genes and 47 un-methylated housekeeping genes as positive
and negative controls to see which preprocessing method is better at identifying these
two different groups of genes (methylated and non-methylated). At each CpG island,
we fit a 75% quantile regression with the array (or cell line, patients) and probes
as covariates, that is, Map = arraya+probep + errorap, , all error terms are assumed to be independent and distribution free. Both "array"
and "probe" are fixed effects. For each array (cell line or patient), we obtain a
p-value from the quantile regression output to indicate whether there is some hypermethylation
signal at 75% quantile for each array. The methylation score given to each CpG island
is the count of the number of cell lines with p-value less than a certain threshold
p0, where we let p0 = 0.05, 0.04, 0.03, 0.02 and 0.01. At each p-value threshold p0, we have an integer methylation score n for each CpG island. The range of n is from
0 to N, where N is 40 and 26 for breast cancer cell line data and ovarian cancer data
respectively. There are in total of Nm and NHK methylation scores for known methylated genes/CpG islands (Nm = 30 for breast cancer data, Nm = 32 for ovarian cancer) and housekeeping genes (NHK = 47) respectively.

In order to see if known methylated genes and housekeeping genes are identified correctly,
we use two different statistical measurements for known methylated genes and housekeeping
genes. One is the statistics of mean difference of methylation scores of two groups
of genes divided by their variance. That is, , where , , and are the mean and variance of methylation scores for known methylated genes and housekeeping
genes respectively, we call this measurement "T.stat". Another measurement is the
area under the Receiver Operating Characteristic (ROC) curve, and we call it "AUC".
For each preprocessing method, the AUC is calculated according to the false positive
and true positive rates defined in the following way. At each methylation level C0 that ranges from 0 to N, the false positive rate is the ratio of the number of un-methylated
housekeeping genes/CpG islands with methylation scores greater than or equal to C0 and the total number of housekeeping genes (NHK), and the true positive rate is the ratio of the number of known methylated genes/CpG
islands with methylation scores greater than or equal to C0 and the total number of known methylated genes (Nm). For both T.stat and AUC, the larger a statistical measurement is, the better a
processing method is.

Comparison results

For each of the two data sets, we choose five p-value cutoffs, 0.05, 0.04, 0.03, 0.02,
and 0.01. At each p value cutoff point, we calculate the two statistical measurements:
"T.stat", that is, mean difference of methylation scores of two groups divided by
their variance, and "AUC", that is, the AUC of a ROC curve, for each of the 20 preprocessing
methods. The results are listed in Tables 2, 3, 4 and 5 for breast cancer cell lines (Tables 2 and 3) and ovarian patients data (Tables 4 and 5). In each of these four tables, there are 3 underlined colored bold numbers indicating
the 3 largest scores of 20 different processing methods. In Table 2, that is, the T.stat measurement of breast cancer data, 13 out of 15 underlined bold
numbers belong to the control LOESS method. The other two belong to the composite
LOESS method. In Table 3, that is, the AUC measurement of breast cancer data, 10 out of 15 bold numbers belong
to the control LOESS method; the other 5 belong to the composite LOESS method. In
Table 4, that is, the T.stat measurement of ovarian cancer data, 10 out of 15 bold numbers
belong to the control LOESS method. The other five belong to the LOESS method. In
Table 5, that is, the AUC measurement of ovarian cancer data, 6 out of 16 bold numbers belong
to the control LOESS method; the other 10 belong to the LOESS method. These summaries
mean that control LOESS is a relatively better normalization method in all 3 tables
except in Table 5, that is, the AUC measurement for ovarian cancer data. Note that there are 16 rather
than 15 underlined bold numbers in Table 5 because there are two measurements that are tied for the third largest numbers. In
addition, tables 2 and 3 show control LOESS normalization without background correction is slightly better
in breast cancer cell line data, while tables 4 and 5 show that the combination of background subtraction and control LOESS work slightly
better than the others.

In order to further compare the performances of different normalization and background
correction methods, at each p-value cutoff point we calculate the average of each
statistical measurement for each normalization method (across five different background
correction methods) and for each background correction method (across four different
normalization methods). The average scores are plotted in Figure 2 (for breast cancer data) and Figure 3 (for Ovarian cancer data). In each of these two figures, two plots in the top panel
are used to compare four different normalization methods using measurements T.stat
and AUC. Two plots in the bottom panel are used for the comparisons of five different
background correction methods using measurements T.stat and AUC. Both Figures 2 and 3 show that there are more differences among normalization methods than among background
correction methods. If we ignore the LOESS normalization method, that is, the blue
line in plots A and C of Figures 2 and 3, we can see that the performance of the other three normalization methods can be
ranked in the following order in both breast cancer and ovarian cancer data: control
LOESS (red curve) is better than composite LOESS which is better than "none" (i.e.,
without any normalization). The global LOESS normalization method is less efficient
than the composite LOESS method in breast cancer data, but it is better than the composite
LOESS method in ovarian cancer data, in which control LOESS and global LOESS have
similar performance. Plots B and D in both Figures 2 and 3 show that the difference between different background correction methods is not very
much, their difference is much smaller than the differences among four normalization
methods.

Figure 2.Breast cancer mean differences of different normalization and background correction
methods. The two plots in the top panel are the results of comparing four normalization methods
using two statistical measurements. The two plots in the bottom panel are the results
of comparing five background correction methods using two statistical measurements.

Figure 3.Ovarian cancer mean differences of different normalization and background correction
methods. The two plots in the top panel are the results of comparing four normalization methods
using two statistical measurements. The two plots in the bottom panel are the results
of comparing five background correction methods using two statistical measurements.

Conclusions and Discussion

In this paper, we compare four normalization and five background correction methods.
There are more differences among normalization methods than background correction
methods. Among four normalization methods, the result of no normalization performs
the worst in that both statistical measurement scores are the smallest in both data
sets. Therefore, it is necessary to do normalization. The control LOESS and composite
LOESS normalization methods provide relatively stable results in both data sets when
the p value threshold changes. However, the LOESS normalization results are more variable
across different p-value cutoff points. On the other hand, the differences among background
correction methods are relatively small. Our comparison results show that even though
some background correction methods are slightly better than others, the differences
are much smaller than the differences among normalization methods. With appropriate
normalization, the need for background-corrected DMH methylation data might be obviated.
This conclusion is consistent with the findings of [8], which are about the gene expression microarray data. That is, differentially expressed
genes are most reliably detected when background is not subtracted. It is also consistent
with the conclusion of [36], which claims that background correction is generally needed to remove bias, but
appropriate normalization obviates the need for mock experiments.

The housekeeping genes used as non-methylated genes are selected from publicly available
data [32] using the following criteria. First, there is one and only one CpG island associated
with this gene. We use this criterion because there could be several CpG islands associated
with one housekeeping gene, in this case we cannot determine the methylation signal
of such housekeeping gene. Second, there are at least three probes and at least one
probe is in the promoter region according to the annotation provided by Agilent. We
use this standard because methylation signals at CpG island with small number of probes
are not reliable according to our previous work [26]. Third, the CpG island associated with a housekeeping gene will have a methylation
score less than or equal to N/2 (that is, half of the number of arrays) in all 20
preprocessing methods and in the data of both breast and ovarian cancer. We choose
housekeeping genes in this way to avoid any bias due to preprocessing methods and
cancer types. Housekeeping genes could have large variability between different samples
and treatments [24,25], especially in cancer tumor or cell lines. For example, some housekeeping genes may
have abnormally high or low expression and/or methylation level in breast cancer but
not in ovarian cancer.

In Table 2 of the ovarian cancer methylation review paper [31], 49 genes are summarized as hypermethylated genes. In our comparisons, we only use
32 of them. The other 17 genes are excluded for one or more of the following reasons:
(1) there is no corresponding CpG island in our methylation microarray data, (2) the
corresponding CpG island has less than 3 probes, (3) the corresponding CpG island
does not cover the promoter or first exon region of this gene according to the annotation
provided by Agilent, or (4) there are several CpG islands corresponding to this gene,
and it is difficult to select one.

The effectiveness of a normalization method depends on whether or not its assumption
is valid. The LOESS normalization assumes that each array has a larger number of probes
(or genes) that are not differentially methylated (expressed), or there is an approximately
equal number of positive and negative log ratios. It also requires that a certain
number of probes with these characteristics should cover a full range of intensities
[37]. However, these assumptions could fail in the breast cancer cell line methylation
data since cell lines usually have more methylation than patients. This might be one
main reason that the control LOESS normalization and composite LOESS normalization
are better (i.e., provide more stable results) than the global LOESS normalization
for some p-value cutoff values in the breast cancer cell line data, but not in the
ovarian cancer data. In addition, copy number variations may occur in cancer patient
tumor and cell lines. This may be one of the reasons that those internal control probes
may have unexpected large or small log ratios. However, it is unlikely that the log-ratios
of all those 199 internal probes will be affected, so copy number variations may not
affect the validity of our results.

In this paper, we did not compare with the Agilent Feature Extraction Software [38] because it has been shown that it does not outperform the LOESS normalization [8]. Although the internal control probes we identified are mainly used for preprocessing
DMH data in this paper, the ideas of our methods can be useful for preprocessing data
generated from other methylation microarray and sequencing protocols [10,11,39,40] that use methylation sensitive or insensitive enzymes to digest DNA fragments.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

SS developed and implemented the models, performed all statistical analyses, drafted
and revised the manuscript. PSY and YWH were involved in the data collection and helped
in preparation of the manuscript. THMH oversaw the project and revised the manuscript.
SL provided suggestions on the project and revised the manuscript. All authors have
read and approved the final document.

Acknowledgements

This work was supported by the National Science Foundation [0112050] while SS was
a postdoctoral researcher in the Mathematical Biosciences Institute, The Ohio State
University. The authors thank Drs. Terry Speed, Greg Singers and Dustin Potter for
valuable suggestions and discussions. In particular, we appreciate that Dr. Potter
shared the methylation-sensitive restriction enzyme cutting sites data (from his previous
DMH publications) with us.